Question

Let \(m_{0}, \ldots, m_{r} \in \mathbb{N}_{\geq 2}\). (i) Prove that every nonnegative integer \(a1\). (ii) Compute the above representation of \(a=42\) for \(m_{0}=2, m_{1}=3, m_{2}=2\), and \(m_{3}=5\). (iii) What is the analogous mixed-radix representation for polynomials?

Short answer

42 has a mixed-radix representation of (0, 0, 1, 3) for bases (2, 3, 2, 5).

Step-by-step solution

Understanding Mixed-Radix Representation

In a mixed-radix system, each digit can have a different base. The task is to express the integer \(a\) as a sum where each term \(a_i\) is multiplied by a product of previous base values \(m_0, m_1, \ldots\). Thus, the problem is similar to creating a number in a base that changes for each positional digit.

Prove Existence of Representation

Given \(a < m_0 \cdots m_r\), express \(a\) through successive divisions:- Set \(b_0 = a\).- For each \(i\) from 0 to \(r\): - Let \(a_i = b_i \mod m_i\), ensuring \(0 \leq a_i < m_i\). - Update \(b_{i+1} = \left\lfloor \frac{b_i}{m_i} \right\rfloor\).- The process stops when \(b_{r+1}=0\).This method always yields unique 'digits' \(a_i\) since \(a < m_0 \cdots m_r\) ensures there's enough room for all coefficients before running out of integer results.

Prove Uniqueness of Representation

Assume two representations exist for \(a\):\[ a = a_0 + a_1 m_0 + \cdots + a_r m_0 \cdots m_{r-1} \]\[ a = b_0 + b_1 m_0 + \cdots + b_r m_0 \cdots m_{r-1} \]Subsequently, this implies that:\[(a_0 - b_0) + (a_1 - b_1) m_0 + \cdots + (a_r - b_r) m_0 \cdots m_{r-1} = 0\]Each \(a_i - b_i\) must be divisible by \(m_i\), forcing \(a_i = b_i\) within range, thus ensuring uniqueness of \(a_i\).

Computing Representation for a=42

Given \(m_0=2\), \(m_1=3\), \(m_2=2\), and \(m_3=5\):Begin with \(b_0 = 42\):- Find \(a_0 = b_0 \mod m_0 = 42 \mod 2 = 0\). Set \(b_1 = \left\lfloor \frac{42}{2} \right\rfloor = 21\).- Find \(a_1 = b_1 \mod m_1 = 21 \mod 3 = 0\). Set \(b_2 = \left\lfloor \frac{21}{3} \right\rfloor = 7\).- Find \(a_2 = b_2 \mod m_2 = 7 \mod 2 = 1\). Set \(b_3 = \left\lfloor \frac{7}{2} \right\rfloor = 3\).- Find \(a_3 = b_3 \mod m_3 = 3 \mod 5 = 3\). Set \(b_4 = \left\lfloor \frac{3}{5} \right\rfloor = 0\).The mixed-radix representation of \(42\) is \((a_0, a_1, a_2, a_3) = (0, 0, 1, 3)\).

Analogous Mixed-Radix Representation for Polynomials

For polynomials, similar to integer representation, consider expressing a polynomial \(P(x)\) as:\[ P(x) = p_0(x) + p_1(x)q_0(x) + p_2(x)q_0(x)q_1(x) + \cdots \]where each \(p_i(x)\) is a polynomial and \(q_i(x)\) serves as the base for each step, analogous to the integer bases \(m_i\). Each \(p_i(x)\) must have a degree less than \(q_i(x)\).

Key concepts

Integer Representation

In mathematics, representing integers in various forms is a fundamental concept. Mixed-radix representation is one such method, where each 'digit' has its own base. This is unlike the conventional decimal system where every digit has the same base (10). The idea is to express any integer as a combination of coefficients with varying bases. Imagine a toolkit with different gears; each one turns at its own speed, contributing to the whole.

For example, consider a number less than the product of several bases \(m_0, m_1, \ldots, m_r\). The task is to express this number in the form \[ a = a_0 + a_1 m_0 + a_2 m_0 m_1 + \cdots + a_r m_0 \cdots m_{r-1} \] where each coefficient \(a_i\) must fall between 0 and \(m_i\). This structure provides a rich framework not only for practical computation but also for deeper mathematical exploration.

Unique Representation Proof

Why can we be confident that every integer has a unique mixed-radix representation? Let's explore the mechanics behind this guarantee. The process starts by ensuring that each coefficient \(a_i\) is uniquely determined (not arbitrary). The proof begins by assuming two supposedly different representations of the same integer. **Mathematical Contradiction**- Consider: \[ a = a_0 + a_1 m_0 + \cdots + a_r m_0 \cdots m_{r-1} = b_0 + b_1 m_0 + \cdots + b_r m_0 \cdots m_{r-1} \] - If both expressions are equal, then \[ (a_0 - b_0) + (a_1 - b_1)m_0 + \ldots + (a_r - b_r)m_0 \cdots m_{r-1} = 0 \] - Since each \(m_i\) is chosen such that it cannot divide \(a_i - b_i\) unless \(a_i = b_i\), this implies that all coefficients must indeed be identical, thus proving uniqueness.The distinct bases assure that overlapping values of coefficients are prevented, providing each number with just one valid mixed-radix form.

Polynomial Representation

In mathematics, polynomials too find themselves in mixed-radix scenarios analogous to integer cases. Imagine extending the mixed-radix concept to polynomial expressions. Rather than using numbers, we use polynomials as units of 'digits' and 'bases'. Consider representing a polynomial \(P(x)\): - It can be expressed as: \[ P(x) = p_0(x) + p_1(x)q_0(x) + p_2(x)q_0(x)q_1(x) + \cdots \] - Each polynomial \(p_i(x)\) takes the role of a 'digit'.- Likewise, \(q_i(x)\) plays the role of the base for each stage of computation. - The degree of each polynomial \(p_i(x)\) must be less than that of \(q_i(x)\).By adhering to these rules, we ensure that polynomials have a unique distribution within their respective constraints, similar to the integrity we require in integer mixed-radix representations.

Step by Step Solution

Let's practically harness mixed-radix representation using a step-by-step guide for clarity. Consider the task of decomposing an integer, such as 42, with the bases \(m_0 = 2, m_1 = 3, m_2 = 2, m_3 = 5\). Here's how you can achieve this:1. **Initialization:** - Start with \(b_0 = 42\). 2. **Iterative Method:** - Calculate each coefficient \(a_i\) using modulus: - \(a_0 = b_0 \mod m_0 = 42 \mod 2 = 0\) - Update \(b_1 = \left\lfloor \frac{b_0}{m_0} \right\rfloor = 21\) - \(a_1 = b_1 \mod m_1 = 21 \mod 3 = 0\) - Update \(b_2 = \left\lfloor \frac{21}{3} \right\rfloor = 7\) - \(a_2 = b_2 \mod m_2 = 7 \mod 2 = 1\) - Update \(b_3 = \left\lfloor \frac{7}{2} \right\rfloor = 3\) - \(a_3 = b_3 \mod m_3 = 3 \mod 5 = 3\) - Final update, \(b_4 = \left\lfloor \frac{3}{5} \right\rfloor = 0\)3. **Result Compilation:** - The mixed-radix representation is \((a_0, a_1, a_2, a_3) = (0, 0, 1, 3)\).

Breaking the task into such manageable steps helps cement understanding and creates a structure that can be applied to any given integer under its constraints.